Università degli Studi del Molise
Facoltà di Economia
Dipartimento di Scienze Economiche, Gestionali e Sociali
Via De Sanctis, I-86100 Campobasso (Italy)
ECONOMICS & STATISTICS DISCUSSION PAPER
No. 55/09
A Characterization of the Dickey-Fuller Distribution,
With Some Extensions to the Multivariate Case
by
Roy Cerqueti
University of Macerata, Dept. of Economic and Financial Institutions
Mauro Costantini
University of Vienna, Dept. of Economics
and
Claudio Lupi
University of Molise, Dept. SEGeS
The Economics & Statistics Discussion Papers are preliminary materials circulated to stimulate discussion
and critical comment. The views expressed in the papers are solely the responsibility of the authors.
A Characterization of the Dickey-Fuller Distribution,
With Some Extensions to the Multivariate Case
Roy Cerqueti∗
Mauro Costantini†
Claudio Lupi‡
Abstract
This paper provides a theoretical functional representation of the density function related to the DickeyFuller random variable. The approach is extended to cover the multivariate case in two special frameworks:
the independence and the perfect correlation of the series.
key words: Dickey-Fuller distribution, unit root
JEL codes: C12, C16, C22
1
Introduction
In this paper we deal with the theoretical case where n possibly cross-dependent time series are
generated by
yi,t = yi,t−1 + ui,t
= yi,0 +
t
X
(i = 1, . . . , n ; t = 1, . . . , T )
ui,j
j=1
= yi,0 + Si,t
(1)
where yi,0 = ci with probability one, or it has a given probability distribution. The ui,t ’s are
assumed to satisfy some regularity conditions so that a suitably normalized transform of Si,t ,
∗ (r) := T −1/2 σ −1 S
Si,T
i,bT rc (where b·c denotes the integer part and r ∈ [0, 1]), is such that
i
∗
Si,T (r) ⇒ Wi (r) as T → ∞, with Wi (r) a Wiener process (see e.g., Phillips, 1987, for a detailed
discussion of such regularity conditions and of the exact meaning of the normalization). Here
we take the simplifying assumption that ui,t ∼ iid(0, σi2 ), so that the conditions for the weak
convergence of the normalized partial sums are trivially satisfied.
It is well known that, under the null H0 : ρi = 1, the t-ratio based on the OLS estimator of ρi , b
tρi ,
in
∆yi,t = ρi yi,t−1 + ei,t .
(2)
∗
(corresponding author) University of Macerata, Dept. of Economic and Financial Institutions, Via Crescimbeni,
20. I-62100 Macerata (Italy). roy.cerqueti@unimc.it
†
University of Vienna, Dept.
of Economics, Bruenner Strasse, 72.
A-1210 Vienna (Austria).
mauro.costantini@univie.ac.at
‡
University of Molise, Dept. of Economics, Management and Social Sciences, Via De Sanctis. I-86100 Campobasso
(Italy). lupi@unimol.it
1
2 The Univariate Case
2
has the non-standard Dickey-Fuller limiting distribution
1
Z
b
tρi ⇒
Z
Wi (r)dWi (r)
0
1
Wi2 (r)dr
− 21
(3)
0
as T → ∞ (see e.g., Phillips, 1987).
Building on Ruben (1962), in Section 2 we derive an explicit formulation of the conventional univariate Dickey-Fuller distribution (3). Although not easy to manage, the proposed formulation can
be used to derive analytical results, as opposed to the conventional simulation approach. In this
respect, our work is related to Abadir (1995). In Section 3 we extend the analysis to the multivariate case by focusing on the special cases where the series are either independent or perfectly
correlated. Section 4 concludes.
In the derivation of the theoretical results, we assume that all the random quantities introduced
in the paper are contained in a filtered probability space (Ω, F, {Ft }t>0 , P ).
2
The Univariate Case
Consider the Dickey-Fuller distribution (3): by definition of stochastic integral, we take a partition
of the interval [0, 1] in N intervals of length 1/N . If N is large enough, we can approximate the
asymptotic distribution of b
tρi under the null as follows:
Z
1
Z
Wi (r)dWi (r)
0
∼
1
Wi2 (r)dr
− 12
∼
0
N
X
!
Wi (k/N ) [Wi (k/N ) − Wi ((k − 1)/N )]
k=1
k=1
=
1
Wi2 (1) − 1
2
!− 12
N
1 X 2
Wi (k/N )
=
N
1
N
N
X
!− 12
Wi2 (k/N )
=: Xi /Yi ,
(4)
k=1
where
Xi :=
Yi :=
1
Wi2 (1) − 1
2
N
1 X 2
Wi (k/N )
N
(5)
! 21
.
(6)
k=1
Therefore, by definition of Wiener process, the distribution of the univariate Dickey-Fuller test
under the stated conditions can be approximated as follows:
1
2 (1) − 1
χ
b
tρi ⇒ Xi /Yi ∼ 2
(7)
1 ,
(CvM0 (1)) 2
where χ2 (1) denotes a Chi-squared distribution with 1 degree of freedom and CvM0 (1) denotes a
zero level Cramér-von Mises distribution with 1 degree of freedom.
2 The Univariate Case
3
In order to derive an analytical expression for the density function of the Dickey-Fuller distribution, we need to introduce the ratio variable Zi := Xi /Yi . The cumulative distribution function of
Zi will be denoted as Fi . For z ∈ R, we have
Fi (z) = P (Zi ≤ z)
= P (Xi /Yi ≤ z)
Z +∞ Z yz
fXi ,Yi (x, y)dxdy ,
=
0
(8)
0
where fXi ,Yi is the joint density function of the random variables Xi and Yi .
The density function fi of the variable Zi is
Z +∞ Z yz
∂
fi (z) =
fXi ,Yi (x, y)dxdy
∂z 0
0
Z +∞
yfXi ,Yi (yz, y)dy .
=
(9)
0
We need to find an explicit form of the density function fXi ,Yi , in order to get an expression for
fi . Fixed x, y ∈ R, we have
fXi ,Yi (x, y) = fYi |Xi (y|Xi = x)fXi (x) ,
(10)
where fYi |Xi is the density function of Yi conditional on Xi , and fXi is the (marginal) density
function of the random variable Xi .
We write the cumulative distribution functions of Xi as:
FXi (x) = P (Xi ≤ x)
1
2
= P
(W (1) − 1) ≤ x
2 i
= P (Wi2 (1) ≤ 2x + 1)
Z 2x+1
= KXi
s−1/2 e−s/2 ds ,
(11)
−∞
where KXi is the normalizing constant. The density function fXi is then
Z 2x+1
∂
fXi (x) = KXi
s−1/2 e−s/2 ds
∂x −∞
2x + 1
−1/2
.
= 2KXi (2x + 1)
exp
2
(12)
2 The Univariate Case
4
The cumulative distribution function of Yi conditional on Xi = x is
FYi |Xi (y|Xi = x) = P (Yi ≤ y|Xi = x)


!1/2
N
X
1
= P 
Wi2 (k/N )
≤ y Wi2 (1) = 2x + 1
N
k=1


"N −1
#!1/2
X
1
= P
Wi2 (k/N ) + 2x + 1
≤ y
N
k=1
!
N
−1
X
2
2
= P
Wi (k/N ) + 2x + 1 ≤ N y
= P
k=1
N
−1
X
!
Wi2 (k/N ) ≤ N y 2 − 2x − 1
.
(13)
k=1
Equation (13) suggests that we need to discuss the distribution of a sum of squared non-independent
zero-mean Gaussian variables.
Denote the Cramér-von Mises distribution as
V :=
N
−1
X
Wi2 (k/N ) ,
(14)
k=1
and
FV (v) = P (V ≤ v) ,
v ∈ R+ .
(15)
Our approach relies on an invariant symmetry property of the random V (see Ruben, 1962). In
particular, Ruben (1962) shows that the cumulative distribution function of V can be written as
series expansions of Chi-squared cumulative distribution functions, i.e. there exists a sequence of
real numbers {λj }j∈N such that
FV (v) =
+∞
X
λj FN −1+2j (v) ,
(16)
j=0
where FN −1+2j is the cumulative distribution function of a Chi-squared random variable with
(N − 1 + 2j) degrees of freedom. By substituting the explicit expression of the F ’s in (16), we
obtain
Z w
+∞
X
FV (v) =
λj
e−s/2 s(N −3+2j)/2 ds ,
(17)
j=0
0
where we assume without loss of generality that the λ’s contain also the normalizing constants
related to the Chi-squared distributions. By (17), then (13) can be written as
!
N
−1
X
FYi |Xi (y|Xi = x) = P
Wi2 (k/N ) ≤ N y 2 − 2x − 1
k=1
=
+∞
X
j=0
Z
λj
0
N y 2 −2x−1
e−s/2 s(N −3+2j)/2 ds .
(18)
3 The Multivariate Dickey-Fuller Distribution
5
The conditional density function fYi |Xi is
+∞
X
fYi |Xi (y|Xi = x) =
j=0
∂
λj
∂y
"Z
#
N y 2 −2x−1
−s/2 (N −3+2j)/2
e
s
ds
0
N y 2 − 2x − 1
×
= 2N y · exp −
2
+∞
X
×
λj [N y 2 − 2x − 1](N −3+2j)/2 .
(19)
j=0
By substituting (12) and (19) into (10), we have
y
N y 2 − 2(2x − 1)
fXi ,Yi (x, y) = 4N K̄Xi √
×
· exp −
2
2x + 1
+∞
X
×
λj [N y 2 − 2x − 1](N −3+2j)/2 .
(20)
j=0
Finally, from (9) and (20) we have
Z
fZi (z) = 4N K̄Xi
+∞
0
×
+∞
X
y2
N y 2 − 2(2yz − 1)
√
· exp −
×
2
2yz + 1
λj [N y 2 − 2yz − 1](N −3+2j)/2 dy .
(21)
j=0
Although rather involved, (21) can in principle be used to derive analytical results on the DickeyFuller distribution.
3
The Multivariate Dickey-Fuller Distribution
This section is devoted to the analysis of the distribution of the multivariate Dickey-Fuller t-ratio.
Let’s define the random vector (Z1 , . . . , Zn ) of asymptotic distributions under the null, accordingly with (3) and (6). More precisely, we can write
1
Wi2 (1) − 1
Zi :=
2
!− 21
N
1 X 2
Wi (k/N )
N
i = 1, . . . , n .
(22)
k=1
Our aim is to provide a closed form expression for the joint density function fZ1 ,...,Zn of the
random vector (Z1 , . . . , Zn ).
Here we study the two extreme cases where the series are either independent or perfectly
correlated.
3.1
Independent Series
Assume that the series y’s are cross sectional independent. Once that the univariate (marginal)
density has been derived, this case becomes trivial. Indeed, for each (z1 , . . . , zn ) ∈ Rn , we can
3 The Multivariate Dickey-Fuller Distribution
6
write the density function fZ1 ,...,Zn simply as
fZ1 ,...,Zn (z1 , . . . , zn ) =
n
Y
fZi (zi ) ,
(23)
i=1
where fZi (zi ) is given by (21).
3.2
Perfectly Correlated Series
Fixed i = 2, . . . , n, we assume that yi and yi−1 are perfectly correlated, and there exists a constant
αi such that
yi = αi yi−1 .
(24)
Of course, the ordering of the series is purely conventional. Any ordering would be possible just
by changing the parameter αi .
The dependence among the variables is reflected on the dependence among the Wiener processes
Wi . In particular, by using the derivation of the Dickey-Fuller asymptotic distribution, then condition (24) can be rewritten in terms of the Wiener processes Wi :
Wi = αi Wi−1 .
(25)
By substituting (25) into (22), we obtain
Zi :=
=
1
2
1
N
(αi Wi−1 (1))2 − 1
1
PN
2
2
2
k=1 (αi Wi−1 (k/N )
|αi | Zi−1 + 1
2
2 (αi
α2i
N
− 1)
1
2 (k/N ) 2
W
k=1
i−1
i = 1, . . . , n .
(26)
PN
Formula (26) allows us to write explicitly the conditional cumulative distribution of Zi given Zi−1 .
Consider zi , zi−1 ∈ R. Then (18) gives
P (Zi ≤ zi |Zi−1 = zi−1 ) =


1
2


2 (αi − 1)
≤
z
= P  |αi |Zi−1 + Zi−1 = zi−1 
i
1
2
2
αi P N
2
k=1 Wi−1 (k/N )
N


"
#2
N
X
2|α
|(z
−
|α
|z
)
i
i
i
i−1
2
.
√
=P
Wi−1
(k/N ) ≤
2 − 1)
N
(α
i
k=1
(27)
Now, consider the joint density function of the random variable (Z1 , . . . , Zn ). Given (z1 , . . . , zn ) ∈
Rn , the dependence condition (25) implies
fZ1 ,...,Zn (z1 , . . . , zn ) = fZ1 (z1 ) ·
n
Y
i=2
fZi |Zi−1 (zi |Zi−1 = zi−1 ) ,
(28)
4 Conclusions
7
where
∂
[P (Zi ≤ zi |Zi−1 = zi−1 )] .
∂zi
Consider i = 1 . . . , n and zi , zi−1 ∈ R. Then (18) gives
 »

–2
2|αi |(zi −|αi |zi−1 )
Z
+∞
√
X 
N (α2 −1)

i
P (Zi ≤ zi |Zi−1 = zi−1 ) =
λj 
e−s/2 s(N −1+2j)/2 ds .
fZi |Zi−1 (zi |Zi−1 = zi−1 ) =
j=0
(29)
(30)
0
The density function of Zi conditional on Zi−1 , fZi |Zi−1 , is then
 »

–2
Z 2|αi√|(zi −|αi |zi−1 )
+∞
X
∂ 
N (α2 −1)

i
e−s/2 s(N −1+2j)/2 ds
fZi |Zi−1 (zi |Zi−1 = zi−1 ) =
λj

∂zi
0
j=0

"
#2 

− |αi |zi−1 )
1 2|αi |(zi − |αi |zi−1 ) 
√
=
×
· exp −
 2

N (αi2 − 1)2
N (αi2 − 1)
"
#N −1+2j
+∞
X
2|αi |(zi − |αi |zi−1 )
√
×
λj
.
(31)
N (αi2 − 1)
j=0
8αi2 (zi
4
Conclusions
In this paper an explicit approximation of the density function of the multivariate Dickey-Fuller
random variable is provided. We proceed by analyzing at first the univariate case. Our result is
grounded on an invariant symmetry property of some random variables involved in the DickeyFuller distribution (see Ruben, 1962).
As in Abadir (1995), the followed approach allows us to avoid the conventional simulationbased approach. The theoretical results regarding the univariate case are then extended to the
multivariate framework under the assumptions of independent and perfectly correlated series.
Although the independent and the perfectly correlated cases are two extreme settings, they
represent the starting point for exploring models with less restrictive assumptions. In this respect,
the analysis of a general cross sectional dependence case is already in our research agenda.
Moreover, while we deal here only with the Dickey-Fuller distribution in the absence of deterministic terms, further extensions are under scrutiny to cope with the “constant” and “constant
plus linear trend” cases.
References
Abadir, K. M. (1995), “The Limiting Distribution of the t Ratio under a Unit Root,” Econometric
Theory, 11, 775–793.
Phillips, P. C. B. (1987), “Towards a Unified Asymptotic Theory for Autoregression,” Biometrika,
74, 535–547.
Ruben, H. (1962), “Probability Content of Regions Under Spherical Normal Distributions, IV:
The Distribution of Homogeneous and Non-Homogeneous Quadratic Functions of Normal Variables,” Annals of Mathematical Statistics, 33, 542–570.